Question

If x^3 + y^3 = 9, find the value of d^2y/dx^2 at the point (2,1)
The value of d^2y/dx^2 attt thhhe point (2,1) is...​

Answer 1

$\huge{\underline{\mathtt{\red{Ꭺ}\red{ղ}\green{Տ}\blue{ω}\purple{Ꭼ}\orange{я᭄}}}}$

$\begin{gathered}{\bf{Given\::}} \\ \end{gathered}$

The equation given as,

$\begin{gathered}\bf{x^3\:+\:y^3\:=\:9} \\ \end{gathered}$

$\begin{gathered} \\ {\bf{To\: Find\::}} \\ \end{gathered}$

$\begin{gathered}\bf{\dfrac{d^2y}{dx^2}\:at\:point\:(2\:,\:1)} \\ \end{gathered}$

$\begin{gathered} \\ {\bf{Solution\::}} \\ \end{gathered}$

$\begin{gathered} \\ \bf \implies x^3 + y^3 = 9\\ \end{gathered}$

$\begin{gathered} \\ \bf \implies y^3 = 9 - {x}^{3} \\ \end{gathered}$

$\begin{gathered} \\ \bf \implies y = (9 - {x}^{3})^ \frac{1}{3} \\ \end{gathered}$

• Differentiate with respect to 'x' –

$\begin{gathered} \\ \bf \implies \dfrac{dy}{dx} = \dfrac{1}{3}(9 - {x}^{3})^ {- \frac{2}{3}} \times ( - 3 {x}^{2} ) \\ \end{gathered}$

$\begin{gathered} \\ \bf \implies \dfrac{dy}{dx} = - {x}^{2} (9 - {x}^{3})^ {- \frac{2}{3}}\\ \end{gathered}$

• Again Differentiate with respect to 'x' –

$\begin{gathered} \\ \bf \implies \dfrac{d^{2} y}{dx ^{2} } = - \left[{x}^{2} \times - \dfrac{2}{3} \times (9 - {x}^{3})^ {- \frac{5}{3}} \times ( - 3 {x}^{2}) + (2x)(9 - {x}^{3})^ {- \frac{2}{3}}\right]\\ \end{gathered}$

$\begin{gathered} \\ \bf \implies \dfrac{d^{2} y}{dx ^{2} } = - \left[{x}^{2}\times (9 - {x}^{3})^ {- \frac{5}{3}} \times(2{x}^{2}) + (2x)(9 - {x}^{3})^ {- \frac{2}{3}}\right]\\ \end{gathered}$

$\begin{gathered} \\ \bf \implies \dfrac{d^{2} y}{dx ^{2} } = - \left[2{x}^{4}\times (9 - {x}^{3})^ {- \frac{5}{3}}+ (2x)(9 - {x}^{3})^ {- \frac{2}{3}}\right]\\ \end{gathered}$

• At (2,1) –

$\begin{gathered} \\ \bf \implies \left[\dfrac{d^{2} y}{dx ^{2}}\right]_{(2,1)}= - \left[2{(2)}^{4}\times (9 - {2}^{3})^ {- \frac{5}{3}}+ (2 \times 2)(9 - {2}^{3})^ {- \frac{2}{3}}\right]\\ \end{gathered}$

$\begin{gathered} \\ \bf \implies \left[\dfrac{d^{2} y}{dx ^{2}}\right]_{(2,1)}= - \left[2{(16)}\times (9 -8)^ {- \frac{5}{3}}+ (4)(9 -8)^ {- \frac{2}{3}}\right]\\ \end{gathered}$

$\begin{gathered} \\ \bf \implies \left[\dfrac{d^{2} y}{dx ^{2}}\right]_{(2,1)}= - \left[32\times (1)^ {- \frac{5}{3}}+4(1)^ {- \frac{2}{3}}\right]\\ \end{gathered}$

$\begin{gathered} \\ \bf \implies \left[\dfrac{d^{2} y}{dx ^{2}}\right]_{(2,1)}=-(32+4)\\ \end{gathered}$

$\begin{gathered} \\ \implies \purple{ \boxed{ \bf\left(\dfrac{d^{2} y}{dx ^{2}}\right)_{(2,1)}=-36}}\\ \end{gathered}$